Common endpoints of generalized weak contractive mappings via separation theorem with applications

Sirous Moradi,Yeol Je Cho,Somyot Plubtieng,Ali P Farajzadeh

doi:10.1186/s13663-015-0449-4

Abstract

In this paper, first, we give the separation theorem which is an extension of the separation theorem due to Jachymski and Joźwik (J. Math. Anal. Appl. 300:147-159, 2004). Then, by using this and the related results, we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property. This result is an extension and correct version of the main result given by Khojasteh and Rakocevic (Appl. Math. Lett. 25:289-293, 2012).

Highlights

Let (X, d) be a metric space and Pcl,bd(X) be the class of nonempty closed and bounded subsets of X
An element x ∈ X is said to be an endpoint of a multi-valued mapping T if Tx = {x}
In, Moradi and Khojasteh [ ] could improve the result by replacing the weak contraction by a general form of weak contractive and, subsequently, this result was extended by Khojasteh and Rakočević [ ] by introducing the concept of the approximate and common approximate K -boundary strict fixed point property. We show that their result is not correct and so we give the correct form of it applying a new method for its proof, by establishing a separation theorem

Summary

Introduction

Let (X, d) be a metric space and Pcl,bd(X) be the class of nonempty closed and bounded subsets of X. In Section , we prove that the common approximate strict fixed point property and the usual approximate endpoint property are equivalent for generalized weak contraction multi-valued mappings T, S : X −→ Pcl,bd(X). By using the separation theorem obtained in Section and the results in Section , we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property. Two multi-valued mappings T, S : X −→ Pcl,bd(X) are said to have the common approximate K -boundary strict fixed point property if there exists a sequence {xn} ⊂ ∂K , where K is a nonempty subset of X and ∂K is boundary of K , such that lim H.

The following example shows that

Thus we have lim H

We define the mapping ψ

Consider the Urysohn integral equations